91Ó°ÊÓ

For each function that is one-to-one, write an equation for the inverse function in the form \(y=f^{-1}(x)\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$y=3 x-4$$

Evaluate each logarithm in three ways: (a) Use the definition of logarithm in Section 5.3 to find the exact value analytically. (b) Support the result of part (a) by using the change-of-base rule and common logarithms on your calculator. (c) Support the result of part (a) by locating the appropriate point on the graph of the function y=\log _{a} x. $$\log _{16}\left(\frac{1}{8}\right)$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log _{5}(x+2)+\log _{5}(x-2)=1$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log _{2}(x-7)+\log _{2} x=3$$

Solve each equation. $$\left(\frac{3}{5}\right)^{-x}=\left(\frac{9}{25}\right)^{1-5 x}$$

For each function that is one-to-one, write an equation for the inverse function in the form \(y=f^{-1}(x)\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$y=4 x-5$$

The table lists heart disease death rates per \(100,000\) people for selected ages. $$\begin{array}{|l|c|c|c|c|c|} \hline \text { Age } & 30 & 40 & 50 & 60 & 70 \\\ \hline \text { Death Rate } & 8.0 & 29.6 & 92.9 & 246.9 & 635.1\\\\\hline \end{array}$$ (a) Make a scatter diagram of the data in the window \([25,75]\) by \([-100,700]\). (b) Find an exponential function \(f\) that models the data. (c) Estimate the heart disease death rate for people who are 80 years old.

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\ln (47 \times 93)$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log _{7}(4 x)-\log _{7}(x+3)=\log _{7} x$$

For each function that is one-to-one, write an equation for the inverse function in the form \(y=f^{-1}(x)\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$y=x^{3}+1$$